On max-plus finite algebraic eigenvectors

Abstract

In max-plus algebra, eigenvalue problems and characteristic polynomials are defined analogously to those in conventional linear algebra. However, the roots of the max-plus characteristic polynomial are not always max-plus eigenvalues. Algebraic eigenvectors have therefore been proposed as vectors corresponding to the roots of the characteristic polynomial. This paper clarifies necessary and sufficient conditions for the existence of finite algebraic eigenvectors, that is, algebraic eigenvectors with all entries real. Specifically, it is shown that their existence is characterized by the final classes of the associated graph, similarly to the existence of original max-plus finite eigenvectors.